Quantum supergroup structure of (1+1)-dimensional quantum superplane, its dual and its differential calculus

Abstract

We show that the (1+1)-dimensional quantum superplane introduced by Manin is a quantum supergroup, according to the Faddeev-Reshetikhin-Takhtajan approach, when it is extended by the inverse of the bosonic variable. We then give its supermatrix element, its corresponding R-matrix and its Hopf structure. This new point of view allows us, first, to realize its dual Hopf superalgebra starting from postulated initial pairings. Second, we construct a right-invariant differential calculus on it and then deduce the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is a non-cocommutative q-deformed one.